As pointed out by [vernon] at some length, tennis
racquets
are strung with a square weave, but an hexagonal
weave,
if possible, would give some interesting dynamics.

Howevertheless, I think there is a better solution.

Traditionally, catgut is split lengthwise to create the
strings with
which a tennis racquet is strung.

An alternative approach, however, presents itself as soon
as you juxtapose the ideas of tennis, catgut, calamari
and
chain-mail.

Instead of splitting the catgut lengthwise, simply slice it
calamarieqsuely, to give a series of rings. These can
then
be used to create a sheet of catgut chain-mail, which
can
be stretched and fixed over the otherwise inconvenient
hole in the middle of the racquet.

It may be objected that catgut is strongest along its
length, rather than radially. If this proves to be a
problem, then it will necessary to use instead the small
terminal portion of the feline gastrointestinal tract,
which
is admirably adapted to provide radial elasticity.

Finally, I am well aware that synthetic materials are
often
used as a substitute for cat gut. However, I am sure this
is
a very bad idea, since I do not believe that the cat will
be
able to absorb adequate nutrients through a polymer.

The gut comes from sheep, not from cats. It's properly called simply mail or maille; chain-mail is a Victorian perversion (which may itself be a tautology). To make mail, some of the rings must be able to be joined during construction, otherwise construction will be topologically challenging in 3 spacial dimensions. Otherwise a fine idea.

[spidermother] I had neglected the complexities of
linking intact rings, so as not to overburden other
readers. However, you are doubtless aware that
rubber bands can be linked into a 2-D mesh.
Moreover, there are topologies which allow multiple
interlinked rings to be cut from a suitable surface;
it's merely a question of breeding cats with the
appropriate intestinal geometry.

[MaxwellBuchanan], my Idea was more about the weave than about tennis. And, just because it might be difficult to make other things with a hex weave, because the strings need to be taut, that doesn't mean it's impossible. For example, think of "fish-net" stockings. Obviously their strings/threads are taut when worn. So, it should be possible to make something that might be called "chicken-wire" stockings....

[pocmloc], I can see how employing a loose definition of a "hexagon", along with how those stockings appear to be constructed, could lead to them being called "chicken wire stockings. However, I want "regular" hexagons, meaning that all 6 sides are the same length, or as close to the same length as possible. From a distance the openings in those stockings look much more like squares than like hexagons.